文摘
We prove that if a convex body has an absolutely continuous surface area measure, whose density is sufficiently close to a constant function, then the sequence pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302257&_mathId=si1.gif&_user=111111111&_pii=S0022123616302257&_rdoc=1&_issn=00221236&md5=3807b0b3b315896b37164e2ac22f24eb" title="Click to view the MathML source">{Πp>mp>K}pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> of convex bodies converges to the ball with respect to the Banach–Mazur distance, as pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302257&_mathId=si2.gif&_user=111111111&_pii=S0022123616302257&_rdoc=1&_issn=00221236&md5=80a0b208a67afc1857cc66ae99e83090" title="Click to view the MathML source">m→∞pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan>. Here, Π denotes the projection body operator. Our result allows us to show that the ellipsoid is a local solution to the conjectured inequality of Petty and to improve a related inequality of Lutwak.